You can contribute to both a 403(b) and a 401(k) but only if the accounts are under different employer’s plans. You cannot contribute to both types of accounts within the same employer’s plan.
In addition all contributions to any 401(a), 401(k) or 403(b) plan are aggregated as to the annual contribution limit and catch up provisions. This means if you are contributing to two separate plans all of the contributions together cannot be in excess of the annual limits. It is your responsibility to notify your employer if you are contributing to another qualified plan to allow tracking of limits.